Lower bounds for mask polynomials with many cyclotomic divisors

Abstract

Given a nonempty set A ⊂ N ∪{0}, define the mask polynomial A(X) = ∑︁ a∈AXa. Suppose that there are s1, ..., sk ∈ N \{1} such that the cyclotomic polynomials Φs1 , ..., Φsk divide A(X). What is the smallest possible size of A? For k =1, this was answered by Lam and Leung in 2000. Less is known about the case when k ≥ 2; in particular, one may ask whether (similarly to the k =1case) the optimal configurations have a simple “fibered” structure on each scale involved. We prove that this is true in a number of special cases, but false in general, even if further strong structural assumptions are added. Results of this type are expected to have a broad range of applications, including Favard length of product Cantor sets, Fuglede's spectral set conjecture, and the Coven-Meyerowitz conjecture on integer tilings.